REPORT 1191

ON THE DEVELOPMENT OF TURBULENT WAKES

FROM VORTEX STREETS

By ANATOL ROSHKO

California Institute of Technology

REPORT 1191

ON THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS By ANATOL ROSHKO

SUMMARY

E'ake development behind circular cylinders at Reynolds numbersfrom 40 to 20,000 was investigatecl in a low-speed wind tunnel. Standard hot-wire techniques were used to s tudy the velocity jluctuations.

The Reynolds number range of periodic vortex shedding i s divided into two distinct subranges. A t R=4O to 150, called the stable range, regular vortex streets are .formed and n o turbu- lent motion i s developed. The range R=150 to SO0 i s a transition range to a regime called the irregular range, in which turbulent velocity$uctuations accompany the periodic formation of vortices. The turbulence i s initiated by laminar-turbulent transition in the free layers which spring from the separation points o n the cylinder. Th i s transition Jirst occurs in the range R = 150 to 300.

Spectrum and statistical measurements were made to study the velocity$uctuations. In the stable range the vortices decay by viscous diffusion. I n the irregular range the diffusion i s turbulent and the wake becomes fu l ly turbulent in 40 to 50 diameters downstream.

I t was found that in the stable range the vortex street has a periodic spanwise structure.

The dependence of shedding frequency on velocity was success- fu l ly used to measurejlow velocity.

Measurements in the wake of a ring showed that a n annular vortex street i s developed.

INTRODUCTION

I t is always difficult to determine precisely the date ancl author of a discovery or idea. This seems to be the case with the periodic phenomena associated with flow about a cylinder. Although the effect of wind in producing vibra- tions in wires (aeolian tones) had been known for some time, the first experimental observations are due to Strouhal (ref. 1) who showed that the frequency depends on the relative air velocity and not the elastic properties of the wires. Soon after, Rayleigh (1879, refs. 2 and 3) performed similar experiments. His formulation of the Reynolcls number dependence demonstrates his remarliable insight into the problem.

These earliest observations were concerned with the rela- tions between vibration frequency and wind velocity. The periodic nature of the wake was discovered later, although Leonarclo da Vinci in the fifteenth century had already drawn some rather accurate sketches of the vortex formation in the flow behind bluff bodies (ref. 4). However, Leonardo's

drawings show a symmetric row of vortices in the wake. The first modern pictures showing the alternating arrange- ment of vortices in the wake were published by Ahlborn in 1902 (ref. 5); his visualization techniques have been used extensively since then. The importance of this phenomenon, now known as the KBrmBn vortex street, was pointed out by Benard (1908, ref. 6).

I n 1911 KBrmBn gave his famous theory of the vortex street (ref. 7), stimulating a widespread and lasting series of investigations of the subject. For the most part these con- cerned themselves with experimental comparisons of real vortex streets with KBrmBn7s idealized model, calculations on the effects of various disturbances and configurations, and so on. I t can hardly be said that any fundamental advance in the problem has been macle since KBrmBn's stability papers, in which he also clearly outlined the nature of the phenomenon and the unsolved problems. Outstanding per- haps is the problem of the periodic vortex-shedding mecha- nism, for which there is yet no suitable theoretical treatment.

However, the results of the many vortex-street studies, especially the experimental ones, are very useful for further progress in the problem. Attention should be drawn to the work of Fage and his associates (1927, refs. 8 to lo), whose experimental investigations were conducted a t Reynolds numbers well above the ranges examined by most other investigators. Their measurements in the wake close behind a cylinder provide much useful information about the nature of the shedding. More recently Kovasznay (1949, ref. 11) has conducted a hot-wire investigation of the stable vortex street (low Reynolds numbers), to which frequent reference will be macle.

Vortex-street patterns which are stable and well defined for long distances downstream actually occur in only a small range of cylinder Reynolds numbers, from about R=40 to 150, and it is to this range that most of the attention has been given. On the other hand, as is well known, periodic vortex shedding also occurs a t higher Reynolds numbers, up to 10' or more, but the free vortices which move down- stream are quickly obliterated, by turbulent diffusion, and a turbulent wake is established.

The present interest in the vortex street is due to some questions arising from the study of turbulent flow behind cylinders and grids. Such studies are usually made at Reyn- olds numbers for which periodic vortex shedding from the cylinders or grid rods might occur. However, the measure- ments are always talren downstream far enough to insure

I Supersedes N.ZC.4 TN 2913, "On the Development of Turbulent Wakes From Vortex Streets" by Anatol Roshko, 1953.

3 2 3 3 2 8 - 5 L 1

2 REPORT I 191-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

I I I ~ I I 1 1 1 1 5 ~ N ' I I ( N I I ( * rclocity fluctuations are obliterated and ~ \ I ( P I 15 coiilpletely turbulent. There are several 1 1 1 1 I ) O I I 11111 co~~scql~ences of this limitation.

I ~ I I , ~ , t l ~ c energy of the velocity fluctuations is quite low ( . o I I I ~ ~ ~ w ~ wit11 the energy near the cylinder, and especially lo\v compared with tlle dissipation represented by the form tl~.ttg. I n attaining the cleveloped downstream state there is cvitlently not only a rapid redistribution of energy among tlic spectral components but also a large dissipatioll. Sec- ond, the theories wliicli tlcscribc Lliesc downstrt~an~ stages do not relate the flow to the initial conditions except very loosely in terms of diinensionless parameters, anil i t is usually necessary to tleterininc an origin empirically (c. g., mising- length theory or similarity tlieories).

On tlic other hand, tliere is evidence tliat sorne features are permanent, so that they must be tlcterminect near tlic beginning of the motion. Onc sucll feature is tlie lorn-wave- number encl of the spectrum wlliclr (in the tlleory of llomo- geneous turbulence) is invariant.

Another is the random element. I t has been pointecl out by Dryilen (refs. 12 and 13) that in the early stages of tllc decay of isotropic turbulence behind grids tlie bulli of tlic turbulent energy lies in a spectral range wliicli is well ap-

proximated by tile simple function &21 cliaracteristic 1 f B n of certain random processes. Liepmann (ref. 14) has sug- gestecl that such a random process may be found in the shedding of vortices from the grids.

I n short, there has been no description, otlier than very qualitative, of the downstream development of walies xvllich, over a wide range of Reynolds number, exhibit a definite periodicity a t the beginning. The measurements reported here were undertalten to help bridge this gap.

The main results sl~ow tlie clownstrean1 development of the walre, in terins of energy, spectrum, ancl statistical properties. This clevelopment is quite different in two Reynolds number ranges, tlle lower one esterlding from about 40 to 150 and tlic upper, from 300 to 10' (and probably lo5), with a transition range between. Tlle lower range is the region of tlle classic vortex street, stable and regular for a long distance down- stream. Tllc fluctuating energy of the flow has a discrete spcctruin a11i1 sinlply ilccays do\vnstream without transfer of encrgy to otl~er frcqucncies. Irrcgnlar fluctuations arc not developed. I n tlie upper rtmge there is still a predominant (shedding) frequency in the velocity fluctuations near the cylinder, and most of tlic energy is concentratecl a t this frequency; however, some irregularity is already developed, and this corresponcls to a contiriuous spectral distribution of somc of the energy. Dox-nstream, tlic discrete energy, a t tlie slledding frequency, is quiclily dissipated or transferred to otllcr frequencies, so that by 50 diameters tlie walie is cornpletcly turbulent, and tlic energy spectrum of tlle velocity fluctuatiorls appronclles that of isotropic turbulence.

All otlicr features of tlie periodic shedcling and wake phe- rlouncna may be classified as belonging to one or the otller of tlic tn70 ranges. This viewpoint allows some systenlatization ill t l ~ e study of wake development.

I n particularJ i t is felt tile possibilities of the vortex street are by no means exllausted. A study of tlie interaction of periodic fluctuations with a turbulent field seems to be a fruitful approach to the turbulence problem itself. I t is planned to continue the present work along these lines.

From a more immediately practical viewpoint an under- standing of the flow close to a bluff cylincler is important in a t ]cast two problems, namely, structural vibrations in members whiclz themselves shed vortices and structural buffeting

by members placed ill the walres of bluff bodies. hlany of these are most appropriately treated by the statis- tical metllods developeil in tlle tlieories of turbulence and otlrcr ranclom processes (ref. 15). These metllocls are easily extended to include the mixed turbulent-periodic phenomena associated with problems such as the two mentioned above.

Tlic research was collducted a t GALCIT uncler the spon- sorslrip aild with the financial assistance of the National Advisory Committee for Aeronautics, as part of a long-range turbulence study directed by Dr. H. W. Liepmann. His advice and interest tl~roughout the investigation, as well as helpful iliscussions with Dr. Paco Lagerstrom, are gratefully acknowledged.

SYMBOLS

AJB constants a,b major and minor axis, respectively, of corre-

lation ellipse CD drag coefficient CDP form clrag coefficient

C= dN2/Nl D outside diameter of ring d cylinder dimension d' distance between free vortex layers d, diameter of ring-supporting wire E wake energy El,E2 components of wake energy due to periodic

fluctuatiolis F dimensionless frequency, n1d2/v F(n) energy spectrum Fl(n),F2(n) energy spectra of discrete energy Fr(n) continuous energy spectrum G(n,) output of wave analyzer a t setting n~ h lateral spacing of vortices h' initial lateral spacing of vortices h* lateral spacing between positions of u', K const ant k integer I; scale I;, scale corresponding to R, 1 downstream spacing of vortices ,Mlb moment, of order k, of probability density Nk "absolute" moment of probabilitg density nl shedding frequency n2=2nl P(() probability clistribution function p ( ( ) probability density (3 area uncler response characteristic

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS 3

tangential velocity in vortex

Reynolds number response characteristic of wave analyzer Reynolds number based on ring diameter time correlation function space correlation function distance from vortex center raclius of vortex Strouhal number, based on cylinder dimension,

n,dlU, Strouhal number, based on distance between

free vortex layers, n,d'/ LT, time scale time of averaging time local mean velocity in x-direction mean stream velocity mean velocity a t vortex center components of velocity fluctuation periodic velocity fluctuations, a t frequencies

n1 and n2 random velocity fluctuation peak root-mean-square value of velocity fluc-

tuation velocity of vortex relative to fluid reference axes an.cl distance from center of

cylincler flatness factor of probability distribution,

M4 in42 strength (circulation) of a vortex Dirac delta function positive number distance between two points, measured in

z-direction 7

L ?I dimensionless frequency, - n u, e U, x dimensionless "time" in life of vortex, - -

U* d dummy variable liinema tic viscosity a value of u clensi ty skewness of probability distribution, M3/M23/2 time interval dimensionless spectrum, U,F,(n)/L half band width of wave analyzer

GENERAL CONSIDERATIONS

Except for the parameters directly related to the shedding frequency, the quantities measured were essentially those bhat are standard in turbulence investigations (cf. refs. 12 to 14). These are briefly reviewed below with some modifica- tions required to stucly the periodic features.

REFERENCE AXES

The origin of axes is t.aken a t the center of the cylinder (fig. 1); x is measured downstream in the direction of the

,. I"..,...,...U. , Honeycomb-----Precision screen ,Uniform .-Adjustable working : section 17~ :\rtl?; , :,~, LL -"on;

Silk cloth.: 40 ,-*,:?* ;Round section 1'1

----- iq-20"- - -

FIGURE 1.-Sketch of GALCIT 20-inch tunnel.

free-stream velocity, z is measured along the axis of the cylinder, which is perpendicular to the free-stream velocity, and y is measured in the direction perpendicnlar to (x, y) ; that is, y=O is the center plane of the wake. The free- stream velocity- is U, and the local mean velocity in the x-direction is U. The fluctuating velocities in the x-, y-, and z-direction are u, v, ancl w, respectively. The flow is con- sidered to be two dimensional; that is, mean values are the same in all planes z=Constant.

1 SHEDDING FREQUENCY

The shedding frequency is usually expressecl in terms of the dimensionless Strouhal number S=nld/U,, where nl is the shedding frequency (from one side of the cylinder), d is the cylinder diameter, and U, is the free-stream velocity. The Strouhal number S may depend on Reynolds number, geometry, free-stream turbulence level, cylinder roughness, and so forth. The principal geometrical parameter is the cylincler shape (for other than circular cylinders, d is an appropriate dimension). However, cylinder-tunnel con- figurations must be taken into account, for example, blockage and end effects. I n water-channel experiments surface effects may have an influence. Usually the geometrical configuration is fixed, and then S is presented as a function of Reynolds number R.

Instead of Strouhal number it is sometimes convenient to use the dimensionless parameter F=n,r12/v, where v is the kinematic viscosity.

ENERGY

The experiments to be described are concerned mainly with the velocity fluctuation in the wake, and especially with the corresponding energy.

The energy of the velocity fluctuation a t a point in the 1

fluid is - p(u2+v"w3 per unit volume, where (u, v, w) is the 2

fluctuating velocity and the bar denotes an averaging (see the section "Distribution Functions"). I n these experi- ments only the component u was measured, and the term "energy" is used to denote the energy in that component only.

The energy intensity is defined as (u/lT,)'. Since the mean flow is two dimensional, the intensity does not vary in the z-direction. At any downstream position in the wake it varies in the y-direction, normal to the wake. The integral of the intensity over a plane normal to the free stream (per unit span) is called the wake energy E:

2 The term "shedding" is used throughout this report, for convenience; it is not meant to imply anything about the mechanism of the formation of free vortices.

REPORT 11 9 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

displays a predominant frequency (as well as harmonics) which is the shedding frequency. However, except in a small Reynolds number range, the fluctuation has random irregularities "superimposed" on i t ; that is, i t is not purely periodic, in the mathematical sense. However, it is con- venient to speak of the "periodic" and "random" or turbu- lent parts of the f lu~tuat ion.~ The energy may be written

- E= J-mm(2Y 0 (1)

Tlre velocity fluctuation in the wake of a shedding cylinder

where 7 is that portion of the e s r g y contributed by tlre random (turbulent) fluctuation, u12 is contributed by the

For r=0, equations (4) and (7) give

R , ( o ) = ~ ~ F ( ~ ) dn=l

where F(n) is defined as the energy sp9ctrum; that is, F(n) dn is the fraction of the energy in the frequency interval n to n+dn. It is the fraction of energy "per unit frequency," as contrasted with the discrete energy spectrum discussed in the section "Energy."

In studies of isotropic turbulence, a t Reynolds numbers corresponding to those in the present experiments, i t is found that t-he energy spectrum is well represented by the form

sides and nz is prominent there, a t least near the "beginning1' of the wake. Higher harmonics are found to be negligible.)

Equation (2) is a kind of spectral resolution, in which 2 and E2 are the energies a t the specific frequencies nl and n2. This type of resolution is called a discrete, or line, spectrum. But is not a discrete spectral component, for i t is the energy in the turbulent part of the fluctuation and contains "all" frequencies. It has a continuous frequency distribu- tion of energy, for which a slightly different definition of spectrum is appropriate. This is postponed until the follow-

periodic fluctuation a t the shedding frequency n ~ , and 2 corresponds to twice the shedding frequency n2=2n1. (The center of the wake feels the influence of vortices from both

ing section. Corresponding to equation (2), an equation may be writ-

ten for the wake energy E and its turbulent and periodic

or, what amounts to Be same iling, that the correlation function is of the form

components: E=E,+E1+Ez (3)

Of particular interest will be the fraction of discrete energy (El+E2)/E a t various stages of wake development.

CORRELATION FUNCTIONS: SPECTRUM

Definitions.-The time correlation function of the fluctu- ation u(t) is defined by

where 7 is a time interval. The time scale is then defined

The Fourier transform of Rt defines another funct,ion

F ( n ) = 4 l m R 1 ( ~ ) cos 2 rn idr (6)

Then, also

R , ( T ) = L ~ F ( ~ ) cos 2 ~ ' n ~ d ~ (7)

A turbulent fluctuation is an irregular variation, with respect to time, which is charae- terized in particular by its randomness and absence of periodicity (cf. ref. 13, p. 9).

If the normalizing factor K= U,/L is used in equation (lo), L being a characberistic length, then equation (6) gives

which may be conveniently written in terms of the dimen- sionless parameters

cp= UoF(n) IL (12%) and

Then

I t is clear from equations (5) and (10) that L is a length scale related to the time scale by

Equation (I lb) is used as a convenient reference curve to compare the measurements reported below.

Periodic functions.-The energy spectrum F(n) is par- ticularly well suited to turbulent fluctuations, for which the energy is continuously distributed over the fre- quencies. For periodic fluctuations the discrete, or line, spectrum is more appropriate, but in the present "mixed" case it is convenient to write the discrete energy, also, in terms of F(n). This may be done by using the Dirac delta function 6(n). Thus the energy a t the shedding frequency nl is

Then the mixed turbulent-periodic fluctuations in the wake of

THE DEVELOPMENT O F TURBULENT WAKES FROM VORTEX STREETS 5

a shedding cylinder are considered to have an energy spzc- trum which is made up of continuous and discrete parts (cf. eq. (2) and appendix A) :

?LaF(n) dn=?LmF, (n) dn+aLmF1(n) dn+

2 l m ~ ? ( n ) iln (16)

t,hat is.

Space correlation function; phase relations.-The cor- relation function defined in equation (4) clescribes the time correlation. Another correlation function which is useful in the present study is one which relates the velocity fluctua- tions a t two points in the wake, situated on a line parallel to the cylinder. This is defined by

where { is the distance between the two points. The corre- sponding scale is

The function R, should be particularly suited to studying turbulent development. Close to the cylinder i t should reflect the regularity connected with the periodic shedding, especially in a regular, stable vortex street, in which there are no turbulent fluctuations. When there are turbulent fluctuations and, especially, far downstream where there is no more evidence of periodicity, R, should be typical of a turbulent fluid; that is, the correlation should be small for large values of {.

The function R, may be obtained by standard techniques applied to the two signals u(z,t) and u(z+{,t). One well- known visual method is to apply the signals to the vertical and horizontal plates, respectively, of an oscilloscope and to observe the resulting "correlation figures" (or ellipses) on the screen (ref. 16). If the signals u(t) are turbulent fluctua- tions, then the light spot moves irregularly on the screen, forming a light patch which is elliptic in shape. The correla- tion function is given by

DISTRIBUTION FUNCTIONS

Random functions.-The probability density p (4) of a ranclom function u,(t) is defined as the probability of finding u, in the interval (t,t+d<). It may be found by taking the average of observations made on a large number (ensemble) of samples of u,(t), all these observations being made a t the same time t . This is called an ensemble aver- age. If u,(t) is a stationary process, as in the present case, then appeal is made to the ergoclic hypothesis and the en- semble average is replaced by the time average, obtained by making a large number of observations on a single sample of u,(t). The probability density p(4) is the number of times that u, is found in (t,<+d[) divided by the total num- ber of observations made. In practice, time averages are more convenient than ensemble averages. The averaging time T, must bc large enough so that a statistically significant number of observations are made. This imposes no hard- ship; i t is sufficient that T, be large compared with the time scale T. If necessary, the error can be computed.

Experimentally, p(l) may be determined by the principle illustrated below :

where a and b are the major ancl minor axes of the ellipse. If u(z,t) is a periodic function, in both time and space,

then the correlation figure is an elliptical loop (Lissajous figure) whose major and minor axes again give R, according to equation (20). Such a case would exist if the wake had a spanwise periodic structure. Then 112,({) would be periodic. A special case of this is R,(f)=l, as would be expected in a vortex street, provided the vortex filaments are straight and parallel to the cylinder and do not 'Lwobble."

Sketch 1.

The most elementary application of this principle is a graphi- cal one using a photographic trace of u,(t). More conven- iently, electronic counting apparatus is employed (see the section "Statistical Analyzer").

The statistics of u,(t) are usually described in terms of the moments of p([) and certain functions clerivecl from the moments. The moment of order k is defined as

Another useful clefinition is

where NL is equal to iM, for even values of k. If p({) is symmetrical, then MI, is 0 for odd values of k but N, is not.

From the definition of p ( f l i t follows that ~,=f-: p(Q 4= 1; < will be normalized by requiring that M2=1/2, that

I i REPORT 1 19 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

Skewness nl; ,,=- hf2312

- I,. I I t ( % rnc>nn-square value u,L= 112.

'1'ln.c~ useful functions derived from tlre moments arc

AT21/2 &f2112 c= ------ = -- N1 NI (25)

wave result in a large incrcasc in the ltigl~cr moments. I t is intcrcstirlg to stutly tllc rclatioll between probability functions ant1 spct-tra, particularly the case wlicrc most of tlle

The probability dcrlsity of a function which is 1):11.t I \ perioclic ancl partly random is cxpcctctl to clisplaj transition from o~rc type to the other Tlle tendency toxvm.tl tlre random probability tlcnsit,y slroultl be strong. For instance, mndom fliictr~atiolrs in tlrc amplitode of a sirlr

A44 Flatness a=- 34; (27)

Periodic functions.-The above dcfilritious may be ex- teniied to thc case of a periodic f~mction ul(t). Tllc probability density can be completely (lctcl.mi1lctl froln .?

single wave lcngtll of ul(t) ; that IS, i t IS sufficicllt to take T, equal to the period. Tllis complctc a 1)rior.i information is a basic difl'erencc betwccn pcrioclica ant1 r.a~rtlom f ~ ~ ~ l c t t o u s I

If u, ( t ) is measurctl cxperimentallj- then t , ([) in eclriatiorr (21) can also be measured. If ul(t) is given in analytic form then t , ( [ ) may be calculated from eqliatiolr (22). 'I'li~ls the tlistribution dcnsitics for sirnplc \t7avc sllapcs arc casily calculated. Table I gives the probability ticrlsities arid mom(Tkts for tllc triangular Wave, sill(' WaV?, alld Stlllel'c wave. Also included is tltc Gaussian pro1)abilitj- dc11sitj-, which is a standa1.d reference for randorn functiorls

The moments of tllc probabilitj- dcnsitics of tlicsc wave shapes are sllown in figure 2. Tllc mornerlts for the randorn function increase much faster than tllosc for the periodic functions. This results from tllc fact that the lnaxirllllln values of a periodic function arc fixed by its amplitutlc, wllile for a random function d l va111cs are possible.

0 2 4 6 8 10 12 k

FIGURE 2.-4111pllt1tdc d l s ~ ~ i b l t t l o t ~ I I I O I I ~ C I ~ ~ S . --

4 FOI a pe~iodic lunrtion the clfodtc piiilcil)lc m.lr not bc in\olrd, thc rn~rmhlr n\rt- age and the time average ale not thc ~JII IP (UIIICFS the n~cmhris of the rnyrmhlc ha\? rnndonl phase d11Tzienct's) It 15 Lhc iimc .i\rtdpc that 15 computed hcrc, fol c o m l ~ a ~ i - o ~ ~ with the expellmental re\ults, n hich a10 JISO time J\CLA:CS

energy is discrete 1)ut the fluctuation arnplitude is 1.andom.

EXPERIMENTAL DATA

WIND TUNNEL

Tlrc cspc~imcnts nrerc all rnntlc in tllc GATJCIT 20- by 20-incl1 lo~t7-t,lrbulcIlt.e trlrlllel (fig, 1). ~ 1 , ~ turbulence level is about 0,03 rplle ,Trin(l vciocity lnar be varied from abollt cellti,llctcrs secon(l rnilc per l,o,lr) to 1,200 centimeters per secolrtl (25 rniles per liour).

CYLINDERS

Tlle cj-linders used in the experiments varied in diameter from 0.0235 to 0.635 centimeter. hIusic \vire or clrill 7,s used. TIle (iiarnetc>r tolerances are about 0.0002 cellti- meter, The tylintlers spannctl tho tunnel so that the lengtll in all cases was 50 celltimeters (20 incllcs); tllr cJ-lincicrs passed tltronglt the 11-alls and were fastc~recl or~tsidc the tunnel.

RINGS

Sonle stut{ics rlln(lc of tllc flow bcllinil rillgs. Tllcsc \\,ere up of wire, Earll ring was sul l l~or tc t~ in tlrc tullnel by tllrec tllill slll~l,~l.t \virCS, attncollcil to tlrc riIlp circumference at 120' irrt crvals. Table I1 gives tlie tlimcn- siorrs of tllc rings usctl (wl~crc (1 is the wire tliamctcr, I?, the ring diameter, ant1 tl,, the tliarneter of tllc suppo~.t ~virc).

VELOCITY MEASUREMENTS

Velocities liiglicr than about 400 centimeters per scconcl wcrc rncasmctl \\-it11 a pitot tube, calibratctl against a standard. Tltc pressures \yere read on n precision manom- eter to an accuracy of about 0.002 centimeter of alcohol. Velocities lower than 400 centimeters per second were determined from tlre sl~eilding frequency of a reference cylinder (0.635 centimeter), as esplainecl in the section "Use of Shedding Frequencj- for Velocitj- hIeasurements."

Fluctuating velocities \+-ere nreasurcd \t~itll a hot-wire anemometer (1120 mil platinum). Onlj- u(t), the fluctuating velocit-\-. in the flow direction, has been measurecl so far. Tile hot-wire was always parallel to the cylinder.

TRAVERSING MECHANISM

The hot-\\.ire \\-as rnolilrted on a micrometer llcad \vliicli allowed it to be traversed normal to tllc walic arrtl positionetl to 0 001 centimeter. Tlle 11eacl was mounted on a llorizolltal lead screw r;~-llich allowcii tra~rersing in the flov- ilircction, ill the center plalle of tllc tunnel. 7'hc positioning in this direction was accurate to about 0.01 ccntimctcr. The llori- ~ 0 n f a1 lead Screw collltl be t ~ r n e d tllrotlgll 90' to allow traversing parallel to the cj-linclcr, for correlation or pllasc

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS 7

measurements (see the section "Space correlation function; phase relations"). For this purpose, a second micrometer head with hot-wire could be set up in a fixed position along the line of traverse of the first hot-wire. Then correlations could be measured between this point and the movable one.

ELECTRONIC EQUIPMENT

The hot-wire output was amplified by an amplifier pro- vided with compensation up to 10,000 cycles per second. The amplifier output could be observed on an oscilloscope screen or measured on a Hewlett-Paclrard Model 400c vac- uum-tube voltmeter. Values of 2 were obtained by reading the root-mean-square voltage on the voltmeter. (This voltmeter is actually an average-reading meter; i t reads true root-mean-square values only for a sine wave. A few of the indicated root-mean-square values, for turbulent velocity fluctuations, were checked against true root-mean- square values as obtained from the statistical analyzer (see the section "Statistical Analyzer"); these may differ up to 10 percent, depending on the wave shape, but, a t present, no corrections have been made, since the absolute values were not of prime interest.) Usually only relative values of 2 were required, but absolute values could be determined by comparing the voltage with that obtained by placing the hot-wire behind a calibrated grid.

The frequencies of periodic fluctuations were determined by observing Lissajous figures on the oscilloscope; that is, the amplifier output was placed on one set of plates and a known frequency on the other. This reference frequency was taken from a Hewlett-Packard Model 202B audio oscil- lator, which supplied a frequency within 2 percent of that indicated on the dial.

FREQUENCY ANALYZER

Spectra were measured on a Hewlett-Packard Model 300A harmonic wave analyzer. This analyzer has an adjustable band width from 30 to 145 cycles per second (defined in appendix A) and a frequency range from 0 to 16,000 cycles. The output was computed directly from readings of the volt- meter on the analyzer. I t was not felt practicable to read output in the frequency range below 40 cycles; therefore, the continuous spec,trum was extrapolated to zero frequency.

To determine the discrete spectrum in the presence of a continuous background some care was required. I n such cases the analyzer reading gives the sum of the discrete spectral energy and a portion of that in the continuous spectrum, the proportions being determined by the response characteristic of the wave analyzer. The value in the con- tinuous part was determined by interpolation between bands adjacent to the discrete band and subtracted out to give the discrete value, as outlined in more detail in appendix A.

STATISTICAL ANALYZER

The statistical analyzer, designed to obtain probability functions, operates on the principle described in the section

"Distribution Functions;" here u(t) is a voltage signal. A pulse train (fig. 3) is modulated by u(t) and is then fed into a discriminator which "fires" only when the input pulses exceed a certain bias setting, that is, only when u(t)>5. For each such input pulse the discriminator output is a pulse of constant amplitude. The pulses from the discrinimator are counted by a series of electronic decade counters termi- nating in a mechanical counter.

The complete analyzer consists of 10 such discriminator- counter cllannels, each adjusted to count above a different value of 5. I t will be seen that the probability function obtained is the integral of the probability density described in the section "Distribution Functions;" that is,

P(() =Probability that u(t)> 5

It is possible tao rewrite the moments (see the section "Dis- tribution Functions") in terms of p ( 0 , a more convenient form for calculation with this analyzer. These are also shown in table I.

I Turbulence signof

Pulse generotor output

Modulator output

Discriminator output to counter

FIGURE 3.wRepresentative signal sequence for statistical analyzer.

S REPORT 1 1 9 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

\ I I , I I . 1 . , , 1 1 1 1 ) l ~ ~ t t ~ tlctails of the analyzer and computation 1 1 1 , 1 1 1 c t 1 1 - 11111,1 found in references 17 and 18.

RESULTS

SHEDDING FREQUENCY

\ I I I ( Y . St ~.oi~hal's first measurements in 1878 (ref. 1) the I , - 1 1 , I l c r l t I)c~t\vrcn tllc shedding frequency and the velocity I l r t , I ) c ~ c ~ r r of intcrcst to many investigators. Rayleigh (ref. , I I :i) pointed out that the parameter nld/LTo (now called r 1 1 1 5 St roulltll number S) sllould be a function of the Re-molds I I . Since then there have been many measurements of I 111s ~ . ( ' l t b t io~~sllip (ref. 19, p. 570). One of the latest of these 1, I I I ( L rnc~asurement by Kovasznay (ref. 11), whose determina- I 1011 of S(R) covers the range of R from 0 to lo4. Kovasznay r~lw nrade detailed investigations of the vortex-street flow 1):lttcrn a t low Reynolds numbers. He observed that the 5t1.rc.t is developed only at Reynolds numbers above 40 and t l l n t it is stable and regular only a t Reynolds numbers bc~low about 160.

The present measurements of S(R) are given in figures 4 and 5 . Except a t Reynolds numbers between 150 and 300, the

scatter is small, and the measurements agree with those of Kovasznay. The large number of cylinder sizes used results in overlapping ranges of velocity and frequency so that errors in their measurement should be "smeared" out. I t is believed that the best-fit line is accurate to 1 perc,ent.

The measurements are corrected for tunnel bloc,l<age, but no attempt is made to account for end effects. With the cylinder sizes used no systematic variations were detected.

NATURE OF VELOCITY FLUCTUATIONS

I t was observed, as in Kovasznay's work, that a stable, regular vortex street is obtained only in the Reynolds number range from about 40 to 150. The velocity fluctuations in this range, as detected by a hot-wire, are shown on tlie oscillograms in figure 6 , for a Reynolds number of 80. These were taken a t two downstream positions, x/d=6 and 48, and at several values of y/d. (The relative amplitudes are correct a t each value of xld, but the oscillograms for r/d=48 are to a larger scale than those for x/d=6.) The frequencies and amplitudes are quite steady; it is quite easy to determine tlle frequencies from Lissajous figures (see tlie section "Electronic Equipment"), which, of course, are also steady

FIGURE 4.-Strouhal number against Reynolds ~ltttnbcr for circular cylinder.

THE DEVELOPMENT O F TURBULENT WAKES PROM VORTEX STREETS 9

Reynolds number, R

FIGURE 5.-Strouhal number against Reynolds number for circular cylinder.

(a) x/d= 6. (b) x/d=48,

FIGURE 6.-Oscillograms for R= 80 and d= 0.158 centimeter. 323328-55----2

10 REPORT 1191-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

(a) R = 145

FIGURE 7.-Simultaneous oscillograms. d= 0.158 centimeter; z/d= 6; yld = 1 ; l /d= 50.

Another example, a t R= 145, is shown in figure 7(a). (The double signal was obtained for correlation studies and is referred to later in the section "Spanwise Correlation and Phase Measurements." The dotted nature of the trace is due to the method of obtaining two signals on one screen, using an electronic switch.)

At Reynolds numbers between about 150 and 300 there are irregular bursts in the signal. An example is shown in figure 7 (b), a t R=180 and x/d=6. The bursts and irregu- larities become more violent as R increases. I t is rather difficult to determine the frequency. The Lissajous figure is unsteady because of the irregularity, but, in addition, the frequency, as well as i t can be determined, varies a little. This is the reason for the scatter in this Reynolds number range. Two separate plots of S(R) obtained in two different runs are shown in figure 8. They illustrate the erratic behavior of S(R) in this range.

At Reynolds numbers above 300, signals like that in figure 7 (c) were obtained (near the beginning of the wake). This is typical of the velocity fluctuations up to the highest value of R investigated (about 10,000). There are irregularities, but the predominant (shedding) frequency is easy to deter- mine from a Lissajous figure. The Lissajous figure in this case is not a steady loop, as i t is a t R=40 to 150, but neither

is it so capricious as that a t R= 150 to 300, and the matching frequency is quite easily distinguished from the nearby frequencies.

At x/d=48, in this range, all traces of the periodicity have disappeared and the fluctuations are typically turbulent.

R R

(a) d=0.0362 centimeter. (b) d=0.318 centimeter.

FIGURE 8.-Plot of S ( R ) for two single runs.

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS 11

REGULAR AND IRREGULAR VORTEX STREETS

The above observations show that there are three charac- teristic Reynolds number ranges, within the lower end of the shedding range. These will be called as follotvs:

Stable range 40<R< 150 Transition range 150< R<300 Irregular range 300<R< 10,000f

As noted above, the actual limits of these ranges are somewhat in doubt and may depend on configuration, free-stream tur- bulence, and so forth. Also the upper limit of the irregular range is undoubtedly higher than 10,000. (Periodic fluctua- tions in the wake have been observed up to the critical Reynolds number, about 200,000, but the present measure- ments did not extend beyond 10,000.)

In addition to the differences in the nature of the velocity fluctuations, the ranges are characterized by the behavior of the Strouhal number: I n the stable range S(R) is rapidly rising, in the irregular range i t is essentially constant, and in the transition range it is "unstable."

It will be seen in the further results presented below that all phases of the wake development are different in the two ranges, stable and irregular, and that they are indeed two different regimes of periodic wake phenomena.

RELATION OF SHEDDING FREQUENCY TO DRAG

The relation between the Strouhal number S(R) and the drag coefficient CD(R) has often been noted (ref. 19, p. 421). Roughly, rising values of S(R) are accompanied by falling values of CD(R) and vice versa.

The relation to the form drag is even more interesting. The t,otal drag of a cylinder is the sum of two contributions: The skin friction and the normal pressure. At Reynolds numbers in the shedding range the skin-friction drag is "dissipated" mainly in the cylinder boundary layer, while the pressure drag (or form drag) is dissipated in the wake. It may, then, be more significant to relate the shedding frequency to the form drag, both of which are separation phenomena. The R-dependence of the pressure drag coefficient CDp, taken from reference 19, page 425, is shown in figure 5 . It has several interesting features:

(a) CD, is practically constant, a t the value CD,= 1. (b) The minimum point A is a t a value of R close to that

a t which vortex shedding starts. (c) The maximum point B is in the transition range. (d) In the irregular range CDp(R) is almost a "mirror re-

flection" of S(R) . Since the drag coefficient is an "integrated" phenomenon,

it is not expected to display so sharply detailed a dependence on R as does the Strouhal number, but these analogous varia- tions are believed to be closely related to the position of the boundary-layer separation point, to which both the shedding frequency and the pressure drag are quite sensitive.

USE OF SHEDDING FREQUENCY FOR VELOCITY MEASUREMENTS

The remarkable dependence of the shedding frequency on the velocity and the possibilit,~~ of accurately measuring S(R)

make it possible to determine flow velocities from frequency measurements in the wake of a cylinder immersed in the flow. At normal velocities the accuracy is as good as that obtain- able with a conventional manometer, while a t velocities below about 400 centimeters per second it is much better. (For instance, a t a velocity of 50 centimeters per second the manometer reading is only about 0.001 centimeter of alco- hol.) In fact, in determining S(R) in the present experi- ments, this method was used to measure the low velocities by measuring the shedding frequency a t a second reference cyl- inder of large diameter. The self-consistency of this method and the agreement with Kovasznay's results are shown in figure 4.

For velocity measurements it is convenient to plot the frequency-velocity relation in terms of the dimensionless parameter F (see the section "Shedding Frequency") as has been done in figures 9 and 10. The points on these plots were taken from the best-fit line in figure 4. They are well fitted by straight lines

(la) F=0.212R--4.5 50<R<150 (lb) F=0.212R--2.7 300<R<2,000

which correspond to

(2a) 8 ~ 0 . 2 1 2 (1-21.2/R) 50<R<150 (2b) S=0.212 (1- 12.7/R) 300<R<2,000

Line (2b) has been plotted in figure 4 to compare with what is considered the best-fit line. The agreement is better than 1 percent. If line (2b) is extended up to R= 10,000, the maxi- mum error. relative to the best-fit line, is 4 percent.

30

25

20

5 =22

2 I5 k

10

5

0 20 40 60 80 100 120 140 R=U,d/v

FIGURE 9.-Plot of F itgrti~lst R (50<R<140). ~ = 0 . 2 1 2 ~ - 4 . 5 .

12 REPORT 1 19 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

The plot of F(R) is used as follours: The shedding frequency is observed and F=n1d2/v is calculated (V is easily determined) ; R is found on the F(R) plot and the velocity is calculated from R= U ,~ /V .

WAKE ENERGY

From the velocity t.1-aces on the oscilloscope (figs. 6 and 7) i t is clear that in the regular range the fluctuating velocity u(t) is purely periodic, while in the irregular range some of the fluctuations are random. This difference is illustrated in figure 11 which shows the distribution of energy intensity (u/U,)~ across the wake a t two Reynolds numbers, one in the regular range, a t R=150, and one in the irregular range, a t R=500. Only half the wake is shown for each case; the one a t R= 150 is plotted on the left side of the figure and the one for R=500 on the right.

The total energy intensity (u/Zi,)? a t each point was de- termined directly from the reading on the root-mean-square voltmeter (see the section "Electronic Equipment"). The components a t the frequencies n, and nz were determined by passing the signal through the wave analyzer. The curves in each half of fiiure 11 satisfy the equalities

-- The values of (u/U,)', (u,/U,)', and (u2/Uo)? were obtained by measurement (and a t R=150 are self-consistent) whilc (u,/U,)* was obtained by difference. The absolute values in- dicated are somwhat in doubt since the vacuum-tube volt- meter is not a true root-mean-square meter but are believed accurate to about 10 percent.

The particular feature ill~istrated in figure 11 (already ob- vious from the oscillographs) is the absence of turbulent energy a t R= 150 as contrasted with the early appearance of turbulent energy a t R=500. This contrast is typical of the regular and irregular ranges.

The measurements shown were made at 6 diameters down- stream, but the same features exist closer to the cylinder.

(a) R= 150. (b) R=500.

FIGURE 11.-Wake energy. d=O.190 centimeter; x/rl=(i.

DOWNSTREAM WAKE DEVELOPMENT

The downstream development for the case of figrt~.ca I I (but R=500 only) is shown in figure 12. The distrihulioir of total energy intensity (U/U,)~ is shown on the left of t 111t

figure and the discrete energy in tensity (U,/U,)~, a t tllo HI I 1'4 I ding frequency, is shown on the right. Travcrscbs 1% c t r c t

made a t 6, 12, 24, and 48 diameters downstrcaln ' I ' I I I ~ discrete energy decays quite rapidly and is no longer I r r c L r l s t l t

able a t 48 diameters. (Note that the plot of (u,/li,,)' I I I 2 I diameters is shown magnified 10 times, for clarity.) A plot 4,f ( u ~ / U , ) ~ has not been included since it can no lo~lgc I I t ( ,

measured a t even 12 diameters. The distribution of (71 , I 1 , I

may be obtained from these curves by difference. Figure 13 presents the downstream wake develol)rr~c~rrt I I I

another way. The wake energy E was calculated by i 1 1 t c 8 ~ 1 ,I

tion of curves like those in figure 12 (cf. the sec t io~~ " I C t t ergy") ; that is,

Figure 13 is a plot of the energy ratio (El-+ EZ)/E, t lra~ ~ h , I l i t .

ratio of the discrete energy relative to the total cncarg> I n the irregular range the energies were comptl t t.11 t r I I It la

way a t R=500 and 4,000 (two cylinder sizes in cv tc -11 c.trtat.i

and R=2,900 (one cylinder). Figure 13 shows t l r t ~ t lit* decay in all these cases is similar and the wake is c.orr1ltl1~tr4\ turbulent a t 40 to 50 diameters.

The value of x/d for which E,/E becomes zero \ t ~ t s I I I % I I ~ ~ mined for a variety of cylinders, varying in sizes ft.0111 I I I ~

to 1.3 centimeters and a t Reynolds numbers fl.0111 ~ U ( I ( 4 s

10,000. The value was found to lie between 40 r ~ r l t l 50 $11 1111 cases but closer to 40. A precise determination I. t l ~ l l t t ~ i i l ~

(and not important) because of the asymptotic tr l)ltr.oric~li c t b

&/E to zero (E? is already zero a t less than 12 t l i r ~ r r r c ~ t c ~ l ~ i

I n contrastwith this, the stable range (R=50 ; r r ~ c l loo t l r

fig. 13) hzs no devel~pment of turbulence beforc 50 ~ l t r ~ r t l ~ ~ t ( ~ ~ a i

The plots for R= 150 and 200 illustrate the rat1it.r spt.ctt r t c ~ l t l r i r

transition from the stable range to the irregular

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS

(a) Total energy intensizy ( u / U , ) ~ . (b) Discrete energy intensity (u1/U,)2.

Curve for x/d=24 magnified 10 times for clarity.

FIGURE 12.-Wake development. d=0.190 centimeter; R=500.

Downstream distance, diam

FIGURE 13.-Decay of discrete energy.

For R=5O and 100 the energy ratio remained constant a t unity up to x/d=100. Beyond that the energy intensity is so low that the tunnel turbulence cannot be neglected.

MEASUREMENTS OF SPECTRUM

Figure 14 shows spectrum measurements a t 6, 12, 24, and 48 diameters downstream at a Reynolds number of 500, in the irregular range. The lateral position y /d chosen for the measurement a t each x/d is the one for wliich (U,/U,)~ is a maximum (cf. fig. 12). The method of plotting is as follows. The curve through the experimental points is the continuous spectrum F,(n). plotted in normalizecl coordinates. The discrete energies F, = 6(n-n,) and Fz = 6(n- n2) are idicated by narrow "bands" which should have zero widh ?-(I i7finite height but are left "open" in the fi;u,.c. Tic relative energies represented by the areas cn6.c; .iic co- inuos s. curve and under the delta functions, r e s - ~ ~ ' i ~ - l ~ I ~ - f,:c narketl in - -- the figure with values of u,2/u2 and ul-/ u', UZ-/uL.

--- Reference curve tp = 4 1 + (2*a2

FIGURE 14.-Downstream development of spectrum. d = 0.190 centimeter; R= 500; nl=440.

To normalize the continuous spectrum the dimensionless Uo L parameters p=- F,(n) and q=- n are used. In each case L Uo

the curve p= is included for reference. The nor- 1+ (2rrrl)2

malizing coeffic~ent L was determined as follows: (a) Fr(0) was found by extrapolation of the measured

values to n=O. (b) F,(O) and the other values of F,Cn) were normalized

to make ,fF(n)dn= 1. Uo (c) L was found from - F:(O) =4. L.

In short, the measured curve and the reference curve were made to agree in Q,(0) and in area. This requirement deter- mines L.

I n these coordinates the shedding frequency shows an ap- parent increase downstream; this is because the normalizing parameter L increases. For x/d=48 the sheclding frecluency (i. e , n,) is marked with a dash; it contains no discrete energy a t this value of xld.

The "bumps" in the continuous spectrum, near nl and n?, indicate a feeding of energy from the discrete to the continuous spectrum. The portion of the spectrum near n=O, which is established early and which contains a large part of the turbulent energy, seems to be unrelated to the shedding frequency (cf. fig. 15). As the wake develops the energy in the bumps is rapidly redistributed (part of i t

--- Reference curve y, = I + (2r7)'

FIGURE 15.-Downstream developr~le~lt of spectrum. d=0.953 centimeter; R=4,000; nl= 144.

decays) to smooth the spectrum, which, in the fully cle- veloped turbulent wake a t 48 diameters, tends toward the

4 characteristic curve v=-----~. 1 + ( 2 7 4

I n figure 16 the spectrum for x/d= 12 and y/d=0.8 is plotted together with the one a t y/d=O. The curves are sim- ilar a t low frequencies (large eddies) and a t high frequencies; they differ only in the neighborhood of the discrete band. (The slight discrepancy between this figure and fig. 14 is due to the fact that they were measured a t two different times, when the kinematic viscosity v differed. This re- sulted in different values of nl a t the same R.)

A similar downstream development is shown in figure 15 forR=4,000. Here thespectrumatx/d=6issmoother than that in the previous example (fig. 14). This effect may be due not so much to the higher Reynolds number as to the fact that the shedding frequency is closer to the low frequen- cies; that is, the shedding frequency is "embedded" in the low-frequency turbulent band. I t seems to result, a t 48 diameters, in a much closer approach to the reference curve.

Figure 17 shows the spectra a t 48 diameters for ?hree cylinders and several values of yld. It is remarkable that R=4,000, d=0.477 centimeter agrees better with R=500, d=O.190 centimeter than with R=4,000, d=0.953 centi- meter. This seems to bear out the above remark about the relative influence of IZ and n,, for the respective shedding frequencies are 565, 440, and 144.

FIGURE 16.-Spectra at 12 diameters. d=0.190 ( . C ~ I I ( I I I I ~ , I I I 11' ' ) I ~ I I

nl=430.

Finally it may bc noted that values of 16,", \\1111.11 111 1 1 1 ~ 1 1 1 t

11 were obtained by difference, check wcll 1% it 11 I 1 1 1 6 \ J I ~ I J ,

computed from z= ~ ~ , ( n ) d n (before 1101.1111111~11( I O I J t t f

Fr(n)). Spectra for the regular range are not ~~rc~sc~rli~~ll I 11 , \

are simple discrete spectra.

SPANWISE CORRELATION AND PHASE MEAHIIItIa:M &:h 1 :.

The function K, was not measured, but t11(. I I I I L I I I 11 I I I I I I ~

of the spanwise correlation5 are illustrat~cl 111 figlit I- ; 11111~1 I * Figure 7 shows three examples, in rnc.11 of \ \ I I I I 1 1 . I I I I ~ I ~ I P

neous signals were obtained from t w o ]lot-\\ I I I ~ . 111 r .i I ,

and y/d= 1 and separated by 50 dianictl~tr\ \ l l i i r t \ \ I .t ' I lit two signals were obtalned simultaneotlsl 011 I 1 1 1 s (I,, 1 1 1 0 $ 4

screen by means of an electronic su-itc.11 '1'111, 11, I I ~ I I I ~ I 4 f r e t

the dotted trares. At R=145 (fig. 7 (a)) the correlat~oii IS 1 ) c ' r TI . ( I 1 ~ 1 1 1 t l u r t

is a phase shift. At R=180 (fig. 'i ( I ) ) ) 1111, I I I I 1 1 l l t c l ~ t ~ t $ 4

still good, hut the individual sig11:~ls ~ I . I . I I \ I I I I I I I ~ ~ \ Jtit~ik

- In thc remainder Of this section a distinction is 111nrlv l i t , l uvvlt I In 1 3 z r r t ' , ~ s t B , i a a - . b .

function" and "correlation." ?'he former refers to thv 11111~111111 ~ 1 1 . 1 1 1 1 ~ ~ 1 1 1 ~ i t i i _+

"Space correlation function; pllase relations," whilc: Lhn L i t 1 t . r I. #I.I.#I 1 4 1 , I.P..~.. . l . - , . i l

tive sensc.

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS 15

FIGURE 17.-Spectra a t 48 diameters.

4.0

3.5

3.0

2.5

'P

2.0

1.5

Figure 18 sllows the correlation figures obtained by plac- ing the signals of the two hot-wires on the horizontal and vertical plates, respectively, of the oscilloscope.

For N=80 and {/d=100 a steady Lissajous figure is obtained, sliowing that the periodic fluctuations a t the two points (100 diameters apart) are perfectly correlated (but they are not in phase).

For R=220 and 500 there is good correlation only a t small values of {Id, that is, only when the two hot-wires are in the same "eddy," so to spealr. For R=500 the figures are s imil~r to those obtained in fully developed turbulence.

In obtaining these correlations a remarkable phenomenon was observed. Tbe stable vortex street (i. e., R<150) has a perioclic spanwise structure. This was shown by the phase shifts on the Lissajous figure, as the movable hot-wire was traversed ~ara l l e l to the cylinder. From the phase coinci- clences observed, the wave length parallel to the cylinder was about 18 diameters a t a Reynolds number of 80. I t has not been determined whether this periodicity structure is due to a "waviness" in the vortex filaments or whether the vortex filaments are straight but inclined to the cylinder axis. \

STATISTICAL MEASUREMENTS

A few amplitude distribution functions were measured and are slsown in figure 19. One measurement is in the stable range; the other shows downstream development in the irregular range.

n\ D R= 500

3.7 -4 I

7 \ b d l

'\ 0 I

% I d

L

i" 4 B 4

- ;v

k

E

FIGURE 19.-Distribution functions. d=O.190 centimeter.

Tlrc tablc in figure 19 shows values of c and a computed from these curves. The behavior, of course, is as expected,

(dl R =500; (el R = 500; ( f ) R = 500, S/d = 3. </d = 10. </d = 105.

L-84891

3

i \ %F \

FIGURE 18.-Correlation figures. d=0.158 centimeter; x/d= 6; y/d= 1; exposure, f / z second.

--- Reference curve v= ----%-- 1+(2rr?,)~

d=0.953cm Y = , 0 o R=4,000 d 2

dz0.477 Cm O A

R= 4,000 $ = 2 v

3 D

d=0.190 cm Y = o u

down. The breakdowns are uncorrelated a t this distance of 50 diameters. At R=500 (fig. 7 (c)) each signal still shows a predominant frequency. There is some variation in phase between the two signals. The amplitude irregularities appear to be uneorrelated. .

but the numerical values are of some interest. These values (and the curves) show that a t R=100 the signal was prac- tically triangular but had rounded "tops." At R=500 the downstream clevelopment of randomness is shown by the tendency of e and a toward the Gaussian values.

The clistribution is in fact not Gaussian, as may be seen in the figure, for its sliewness u is quite high.

VORTEX RINGS

The flow behind wire rings was briefly investigated. The dimensions of the rings used are given in table 11.

16 REPORT 11 9 1-NATIONAL ADV~SORY COMMITTEE FOR AERONAUTICS

With the rings of diameter ratio D/d= 10 vortices are shed from the wire in almost the same way as from the straight wire, and there is apparently an annular vortex street for some distance downstream. The Strouhal number, meas- ured from R=70 to 500, is lower than that for the straight wire (about 3 percent at R=500 and 6 percent a t K=100).

Fluctuating velocity amplitudes were measured in the wake a t several downstream positions. The results for the largest ring, measured along a diameter, are shown in figure 20. I t should be noted that 43 rather than the energy has been plotted here (cf. fig. 11); only relative values were computed. Close behind the cylinder the wake behind the wire on each side of the ring is similar to that behind the straight wire, but the inside peaks are lower than the outside peaks. This may be partly due to the interference of the hot-wire probe, for a similar effect, much less pronounced, was noticed in the measurements behind a straight wire.

Farther downstream there was some indication of strong interaction between the vortices, for a peak could not be followed "smoothly" downstream. However, the investi- gations were not continued far enough to reach conclusive results. At about 40 diameters downstream the flow became unstable.

Y/D

FIGURE 20.-Shedding from a ring, d=0.168 centimeter; D=1.59 centimeters; R= 100; nl=84. .

They obskrved this to be at about R,=200, with a corre- sponding SD of 0.12.

Again, these experiments were too incomp1e.te to warrant definite conclusions, but the difference in behavior for D/d= 10 and D/d=5 is interesting. This behavior is similar to that observed by Spivack (ref. 21) in his investigation of the frequencies in the wake of a pair of cylinders which were separated, normal to the flow, by a gap. He found that when the gap was just smaller than 1 diameter instability occurred. For larger gaps the cylinders behaved like indi- vidual bodies, while for smaller gaps tthe main frequencies were, roughly, those corresponding to a single bluff body of dimension equal to that of the combined pair, inc,luding the gap.

DISCUSSION

The most significant results of this investigation may be discussed in terms of the Reynolds number ranges defined in the section "Regular and Irregular Vortex Streets," namely, the stable range from R=40 to 150, the transition range from R=150 to 300, and the irregular range above R=300.

STABILITY

The transition range from R=150 to 300 displays the characteristics of a laminar-turbulent transition, and i t is instructive to compare the stability of the flow around the cylinder with boundary-layer stability. The flow in the irregular range has turbulent characteristics, while in the st~able range i t is essentially viscous.

The Reynolds number regimes may be described as follows: Below R=40 the flow around the cylinder is a-symmetric, viscous configuration, with a pair of standing vortices be- hind the cylinder. At about R=40 this symmetric configu- ration becomes unstable. I t changes to a new, stable con- figuration which consists of alternate periodic breaking away of the vortices and formation of a regular vortex street. The change at R=40 is not a laminar-turbulent instability; it divides two different ranges of stable, viscous flow. In either range, disturbances to the stable configuration will be damped out.

On the other hand, the transition range from R= 150 to 300 involves a laminar-turbulent tran~it~ion. To understand how this transition is related to the vortex shedding, it is necessary to know something about the formation of the vortices. Involved in this formation is the circulating

- a sudden increase in S, and at higher Reynolds numbers, in what corresponds to the irregular range, the shedding is similar to that from a straight wire, while in the stable range the shedding is at a much lower frequency. From the observations made it seems likely that in the stable range the ring acts like a disk, sl~edcling the ~ o r t e x loops observed by Stanton and Marshall (ref. 18, p. 578, and ref. 20). Stanton and Marshall do not give their frequency-velocity observa- tions except at the critical RDJ where shedding first starts.

The ring with D]d=5 behaved somewhat differently. The observed frequencies gave values of Strouhal number as shown in table 111. The table shows values of S and R based on the wire diameter, as well as values of S, and R, based on ring diameter. Between R= 153 and 182 there is

motion behind the cylinder as shown in the following sketch. A free vortex layer (the separated boundary layer) springs from each separation point on the cylinder. This free layer and the Fackflow behind the cylinder establish a circulation from which fluid "breairs away" at regular intervals.

THE DEVELOPMENT O F TURBULEP

Tlie laminar-turbulent transition is believed to occur always in t,he free vortex layer; that is, the circulating fluid becomes turbulent before it breaks away. Then each vortex passing downstream is composed of turbulent fluid.

The point in the free vortex layer a t which the transition occurs will depend on the Reynolds number. This transition was actually observed by Schiller and Linlre (ref. 18, p. 555, and ref. 22) whose measurements were made a t cylinder Reynolds numbers from 3,500 to 8,500. The distance to the transition point, measured from the separation point, de- creased from 1.4 diameters to 0.7 diameter, and for a given Reynolds number these distances decreased when the free- stream turbulence was increased. Dryden (ref. 23) observed that a t some value of R, depending on free-stream turbulence and so forth, the transition point in the layer actually reaches the separation point on the cylinder. Transition then re- mains fixed and vortex shedding continues, essentially unchanged, up to Reynolds numbers above 100,000, that is, up to the value of R for which transition begins in the cylinder boundary layer ahead of the separation point. I t is quite likely that even above this critical value of R the phenome- non is essentially unchanged, but now the vortex layers are much nearer together and the vortices are diffused in a much shorter downstream distance.

In summary, vortex formation in the stable range occurs without laminar-turbulent transition. The circulating fluid breaks away periodically, and alternately from the two sides, forming free iiviscous" vortices which move downstream and arrange themselves in the familiar vortex street. In the irregular range transition occurs in the circulating fluid before it breaks away, and the vortices are composed of turbulent fluid. The transition range corresponds to the similar range in boundary-layer stability, and it displays a similar intermittency. The values R= 150 and 300 used to define the range are expected to be different in other experi- ments, depending on wind-tunnel turbulence, cylinder roughness, and so forth.

SHEDDING FREQUENCY

The Strouhal number and Reynolds number dependence is different in the two ranges. In the stable range S(1r') is rapidly rising, while in the irregular range i t is practically constant.

Fage and Johansen, who investigated the structure of the free vortex layers springing from the separation points on various bluff cylinders (ref. 9), made an interesting observe- tion on the relation of the shedding frequency to the distance between the vortex layers. This distance increases as the cylinder becomes more bluff, while the shedding frequency decreases. In fact, if a new Strouhal number Sf is defined in terms of the distance d' between the free vortex layers (instead of the cylinder dimension d), then a universal value S'=0.28 is obtained for a variety of (bluff) cylinder shapes. The measurements of reference 9 were made a t R=20,000, but it is believed that the similarity exists over the whole irregular range. I t does not extend to the stable range. To check this point the shedding frequency was measured in the wake of a half cylinder placed with the flat face broadside to the flow. It was found that S(R) was

WAKES PROM VOI~TEX G'I'IIEE~I'S 17

rising for Itcynoltls nr11111)c~t.s bclow 300 and then became practitanlly c.onslan( : ~ t t l r c , v:iliic S=0.140. For a similar case, at 11= 20,000, L'i~gc :~ntl Jollansen fount1 S=0.143.

The universality of tllc constant Sf is useful in systematiz- ing the shedding phenomena (at least in the irregular range). I t indicates that when the circulating fluid behind the cyl- inder is turbulent then the formation of free vortices is similar for a variety of bluff shapes and over a wide range of Reynolds numbers.

Finally, the relation between St,roullal number and form drag coefficient has been mentioned in the section "Relation of Shedding Frequency to Drag." In the irregular range the slight variations in S(R) reflect slight variations of C,, and so, probably, of the separation point. However, con- stancy of C,, is not enough to insure a fixed separation point. For instance, Cb, remains practically constant down to Reynolds numbers below the shedding range, but the separa- tion point there is farther back than it is a t higher Reynolds numbers. I t would seem worth while, and fairly easy, to measure the position of separation as a function of Reynolds number over the wl~ole shedding range, that is, to complete the data available in the literature.

DOWNSTREAM DEVELOPMENT

Tile way in which the wake develops downstream is quite , different in the stable and irregular ranges.

When the circulating fluid breaks away before the occur- rence of transition in the free vortex layers (i. e., below R= 150)) then the free vortices which are formed are the typical viscous vortices. There is no further possibility for the fluid in them to become turbulent. The vortices simply decay by viscous diffusion as they move downstream (see the section "Spread of Vortex Street" in appendix B).

When turbulent transition does occur, then the vortices which are formed consist of turbulent fluid. They diffuse rapidly as thcy move clomnstrcam and are soon obliterated, so that no evidence of the shedding frequency remains. This development to a completely turbulent wake takes place in less than 50 diamctcrs. In terms of the decay of the discrete energy (fig. 13), the development is roughly the same for Reynolds numbers from 300 to 10,000. This again indicates a remarkable similarity over the whole irregular range.

The stable ancl irregular ranges are also characterized by the difference in the energy spectra of the velocity fluctua- tions. I t has been pointed out that in the irregular range a continuous, or turbulent, part of the spectrum is established at the beginning of the wake development. This turbulence is a result of the transition in the free vortes layers and might be expected to be independent (at first) of the periodic part of the fluctuation, which results from the periodic shedding. Indeed, most of the energy at first is concentrated at the shedding frequency nl (some at na), and it may be represented as a discrete (delta function) part of the spectrum, within the accuracy of the measurements (cf. appendix A). However, the continuous and discrete parts are not entirely independ- ent, as shown by the bumps near nl and nz (fig. 14). This may be regarded as a result of energy "feedingJ' from the discrete to the continuous parts of the spectrum, and i t proceeds in a way which tends to smooth the spectrum. Such

18 REPORT I 19 I-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

transfer of energy between spectral bands is a process de- pending on the nonlinear terms of the equations of motion. The "activity" in the spectrum, a t any stage of its develop- ment, may be regarded as an equilibrium between t,he nonlinear and the viscous terms. I t is an important problem

the speed of rotation this laminar, periodic structure be-

comes unstable and the fluid becomes turbulent, but alternate ring-shaped vortices still exist a t speeds several hundred times the critical speed (ref. 25).

STATISTICS

in the theory of isotropic turbulence. The spectral activity near the frequency of discrete energy

might be loolred upon as a simplified case in which a single band has an excess of energy and the spectral energy flow is unidirectional, that is, out of it into the adjacent bancls. However, the nonhomogeneous character of the field involved (the make) reduces the simplicity, for it is necessary to talce account of energy transfer across the wake. One interesting possibility is to superimpose a homogeneous (isotropic) turbulent field, by means of a screen ahead of '1" shedding cylinder, and to study the effect of this field on the spectral activity near the discrete band. Al tho~gh the wake will still introdme nonhomogeneity (not even counting the periodic part of the motion), it may be possible to arrange the relative magnitudes to give significant results from the simplified model.

To study such problems the technique for measuring the spectrum (appendix A) the of discrete energy will be improved.

To summarize, i t is suggested that the initial development of the spectrum might be regarded as follows: The continu- ous and the discrete parts are established independently, the one by the transition in the vortex layers and the other by the periodic shedding. The turbulence due to the transition is the "primary" turbulent field and its spectrum is the typical, continuous (turbulent) spectrum. (It has been noted in the section "Measurements of Spectrum" that the low- frequency end of the spectrum is established early; it would contain only energy of the primary field.6) The discrete part of the spectrum is embedded in the turbulent part, and it thereby is "excited" into spectral transfer. Some of its energy is transferred to the adjacent frequency bands result- ing, initially, in the development of bumps in the continuous spectrum. Subsequently, as the spectral transfer proceeds, the spectrum becomes smooth.

The above discussion is an abstract way of saying that the vortices are diffused by a turbulent fluid (instead of a viscou~s one). The diffusion involves the nonlinear processes typical of turbulence; the study of these processes, in terms of spectrum, is an important problem.

There is a similar case of turbulent, periodic structure in the flow field between two cylinders, one of which rotates. Taylor's discovery of the periodic structure of the flow is well known (ref. 24). When the inner cylinder rotates, it is possible to obtain a steady, regular arrangement of ring vortices, enclosing the inner cylinder, and having, alternately, opposite directions of circulation. Above a critical value of

6 In the theory of homogeneous turbulenc~ it 1s shown that the low-lrequencv pnd of the spectrum is ~nvarlant, a property related to the I,olts~anshl mlarlnnt

The probability distribution functions (fig. 19) display the characteristics which are expected, from the other observa- tions. The contrast between the functions a t R=100 and R=500, that is, in the stable and irregular ranges, respectively, is quite evident. In the irregular range, even at 2/d=6, where most of the enern is discrete, there is a marked irreplarity in the fluct,lation, as shown by the high value of a.

However, these descriptions are little better than qualita- tive, and it is hoped to obtain more interesting results by extending these statistical methods. of interest in the development of random from periodic motion would be the relation the probability distributions and the the spectra, F~~ instance, it is plain that a purely periodic function (discrete spectrum) will have probability dis- tribution with finite cutoff, while development of random irregularities in the function's amplitude is strongly re- flected in (I) a of the distribution function to higher values of < and (2) the appearance of a continuous spectrum. However, the relation between the two is not unique; that is, the spectrum does not give (complete) information about the probability distribution, and vice versa.

SUGGESTIONS FOR FUTURE INVESTIGATIONS

Some further lines of investigation indicated by these experiments are summarized below.

(a) The transition from the stable to the irregular range should be investigated with controlled disturbances, for example, cylinder roughness and free-stream turbulence. I t is expected that the limits of the transition range (roughly R=150 to 300 for the experimental conditions here) will be lower for higher free-stream turbulence or cylinder roughness. The critical cylinder Reynolds numbers should be related to corresponding numbers for the transition point in the free vortex layers (based on distance from separation point or on the thickness of the layer).

Such studies of stability to different disturbance amplitudes and frequencies are well lrnown in the case of the boundary layer. A variation of the experiments of Schubauer and Skramstad (ref. 26), who used an oscillating wire in the boundary layer to produce disturbances of definite fre- quencies, would be to use a second shedding cylinder.

(b) A study of the spectral development in the neighbor- hood of a discrete band, the effect of a turbulent field on its activity, and so forth (discussed in the section "Downstream Development") may be the most fruitful continuation of ,

these experiments. So far, the problem has been approached only in the theory of isotropic turbulence, where it has not advanced much beyond the similarity considerations of

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS 19

Kolmogoroff, and very little is known about the form of the spectral transfer function.

Interactions between discrete bands, for example, a t slightly different frequencies, can be stuclietl by the use of two or more cylinders arranged to "interfereJ' with each other (some such studies have been macle by Spivacli (ref. 21) but not from this viewpoint), or possibly by using one cylinder having diameter changes along its span.

(c) Townsend has recently used the concepts of intermit- tently turbulent flow and local isotropy in his investigations of the turbulent walie and has obtained a new description of its structure (ref. 27). His studies were made a t downstream distances of 80 diameters or more, so that the walie was fully turbulent. Probably the structure he describes is essentially the same up to the beginning of the fully developed walic (about 50 diameters), but then there is the question of how i t is related to the earlier developments. Tlle most obvious "early developments" are the turbulent transition in the free vortex layers and the periodic sheclding. (Although the shedding frequency is no longer distinguished far clown- stream, it is prominent in the early spectral developments and thus has an influence on the downstream walic.)

Such studies will involve considerably more detailed investigations of the wake structure tllan wcrc made here, possibly along the lines of Townsend's experiments and the classical measurements of energy balance across the walie. The other two components of the energy 7 and 3 will bc needed.

(cl) The nature of the circulating flow behind the cylincler ancl the formation of free vortices, that is, the shedding mechanism, should receive further attention.

(e) The spanwise periodic structure of the vortex street should be investigated, beyond the very cursory observations made here. In particular, a study of the stability of single vortex filaments seems important.

(f) Measurements of the fluctuating forces on the cylilincler, due to the shedding, would be interesting and should have

might be obtained either by direct measurement of forces (on a segment) or pressures (with pressure pickups) or inferred from measuroments of the velocity fluctuations close to the cylincler. In addition to the magnitude of the force or pressure fluctuations, their spanwise correlation is of prime importance.

CONCLUSIONS

An experimental investigation of the walie developed behind circular cylinders a t Reynolds numbers from 40 to 10,000 indicated the following conclusions:

1. Periodic walie phenomena behind bluff cylinders may be classifiecl into two distinct Reynolds number ranges (joined by a transition range). For a circular cylincler these are :

Stable range 40<R<150 Transition range 150<R<300 Irregular range 300<R< 10,000+

In the stable range the classical, stable Kilrmiln streets are formccl; in the irregular range the periodic shedding is accompanied by irregular, or turbulent, velocity fluctuations.

2. Tllc irregular velocity fluctuation is initiated by a laminar-turbulent transition in the free vortex layers which spring from the separation points on the cylinder. The first turbulent bursts occur in the transition range defined abovc.

3. In tllc stable range the free vortices, which move clownstresm, decay by viscous diffusion, and no turbulent motion is developed. In tho irregular range the free vortices contain turbulent fluid and diffuse faster; the wake becomes fully turbulent in 40 to 50 diameters.

4. A velocity meter based on the relation between velocity and slledding frequency is practical.

5. In the stable range a spanwise periodic structure of the vortex street has been observed.

6. An annular vortex-street structure has been observed behind rings having a diameter ratio as low as 10.

immediate practical applications. Thcre seems to be very CALIFORNIA INSTITUTE OF TECHNOLOGY, little information about the magnitude of these forces. I t PASADENA, CALIF., May 29, 1952.

APPENDIX A EXPERIMENTAL ANALYSIS OF SPECTRUM

These note,s supplement th: brief descriptions in tlre I sections ''Frequency AnalyzerJ' and l'Measurcments of Spectrum.'

ANALYZER RESPONSE

Consider the response of a spectrum analyzer, such as that used in tile present experiments, to a mixed pedodic- random input, and in particular consider the problenl of inferring the input from the output.

The input, an energy or power, has a ranclom ancl a periodic component :

The corresponcling spectra are defined by

20 REPORT 11 9 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

APPENDIX B NOTES ON VORTEX-STREET GEOMETRY AND SHEDDING FREQUENCY

where Fl(n) =6(n-n,) (A3)

and 6(n) is the Dirac delta function. The response ~haracteristic of the analyzer may be ob-

tained by considering the effect of a periodic input. When bhe analyzer setting nA coincides with the input frequency n1 the output i3 a maximum, and when the setting is moved away from nl the output falls off. The response character- istic is

Output a t setting nA R(nl-n~)'~utput at setting n,=nl

=R(n,-nl) (A4)

The output spectrum G(n,) of the analyzer is related to the input spectrum F(n) by (cf. ref. 15)

a(n,)=Sm F(n)R(n-n,)dn o

- - ="im F,(n)R(n-n,)dn+& 6(nl-n,)R(n-nA)dn " '1- uZ u2 - -

ur2 - u

(A5) =- F.(n)R(n-n,)dn+-R(n~-nA) u2 u2

Since R(n-n,) is sharp, that is, almost a delta function (see the section "Half Band Width"), F,(n) may be considered to be constant over the significant interval of integration in equation (AS). Then

- - u u

G(n,)=+ Fr(n,) &+A R (nl-n,) u2

(A6) U

where

Q=Sm ~ ( n - n A ) d n = R(n-nA)dnA 0 I- (A7)

is the area under the response characteristic. Equation (A6) gives the output for a mixed periodic-

random input. I t is required to find the separate terms which make up this sum. The procedure is outlined in the section "Separation of Discrete Energy" below.

HALF BAND WIDTH

The of the analyzer is determined by its half band width w. This is defined as the number of "cycles off resonance" a t which the output falls off to 0.01 percent; that is

R(nl-w) =0.0001 ( ~ g )

For an ideal analyzer the response characteristic would be a delta function, but even with half band widths from 30 to 145 (which is the range of the analyzer used here) the char-

acteristic is quite sharp, relative to the frequency intervals of interest. The values 30 to 145 seem quite high, but they are a little misleading because of the high attenuation used to define w. For example, if the response-characteristic half band width is 30 cycles per second, i t has a total width of only cycles per second at attenuation.

SEPARATION OF DISCRETE ENERGY

To separate the discrete energy from the continuous spectrum the following procedure is used.

- u2~(nA)0

"I-W nl n , + ~ "a

Sketch 3.

At n1 + w and nl-w (see sketch) the contribution from 2 is only 0.01 percent, so the measured points there are assumed to lie on the continuous spectrum. It is assumed at first that the continuous spectrum between these points may be deter- mined by interpolation, and its value a t n, is calculated. Then 3 is determined by difference and the last term in equation (87) is calculated, since the form R(n) is known. The first term in equation (A6) then gives the values of G(nA) in the vicinity of nl ; these should check the measured values.

If , however, the continuous spectrum within the band width has a bump, then the above calculation is not self- consistent, and the true values can be determined by successive estimates of

I n principle the method is satisfactory, but in practice the accuracy is low because in the rsgions of interest, that is, near peak frequencies, it depends on the differences of reliltively large quantities. One of these, R(n), is known precisely, but the precision is difficult to realize since the settings on the analyzer cannot be read accurately enough. For the spectral investigations discussed in the section "Downstream Development" the technique will be improved, by monitoring the analyzer with a counter.

The regularity of the vortex shedding and its sensitivity to velocity changes have undoubtedly intrigued everyone who has investigated the flow past bluff bodies. However, as KBrmBn pointed out in his first papers on the vortex street,

the problem is inherently difficult, involving as it does the separation of the boundary layer from the cylinder, and there is yet no adequate theoretical treatment of the mech- anism.

THE DEVELOPMENT O F TURBULENT WAKES FROM VORTEX STREETS 2 1

The following notes may be useful as a summary of tllc intercsting features of the problem. Thev are based largely on the literature but include some results obtained during the present experiments. Chapter XI11 of reference 19 has a very useful review and list of references.

Klirmtin's theory treats a double row of potential vortices, infinite in both directions. The distance between the rows h and the spacing of the vortices in each row Z are constants. The vortices have strength (circulation) r which, with the geometry, determines the velocity T- of the street relative to the fluid. The theory shows that the configura- tion is stable when the rows are staggered by a half wave length ancl the spacing ratio is

The circulation and velocity relative to the fluid arc then related by

Two of the parameters (h, 1, r, and T7) must be determined from some other consic1erations. I11 the real vortex street they must be related to the conditions a t the cylinder.

REALVORTEXSTREET

The real vortex street, even in the stable range, clifl'ers from the idealized one in the following points:

(1) The street is not infinite. I t starts shortly down- stream of the cylinder and eventually loses its identity far downstream. However, the classical vortex-street patterns extending for 10 or more wave lengths should be a good approximation.

(2) The vortex spacing is not constant. I n particular, the lateral spacing h increases downstream.

(3) The real vortices must have cores of finite radius. These grow downstream, so that the vortices diffuse into each other and decrease their circulation. For the same reason the velocity T' is expected to differ considerably from the theoretical value, since i t is strongly dependent on the configuration.

Related to these considerations is the way in which the vortices are first fornled. At Reynolds numbers below the shedcling range a symmetrical pair of eddies is formed at the bacli of the cylincler. As the Reynolds number increases these two eddies grow and become more and more elongated in the flow direction, until the configuration is no longer stable and become asymmetric. Once this occurs the circulating fluitl breaks away alternately from each side to form free vortices which flow downstream and arrange themselves into the regular, stable vortex street.

7 Possibly the breaking away should be regarded as primary. reallting in asymmetry.

In the irregular ~xngc the process is similar, except that the fluid is turbulent (bccausc of the transition in the free vort es la>-rrs).

DOWNSTREAM VORTEX SPACING

In the flow past a stationary cyliniler the frequency with which vortices of one row pass any point is given by

This must be the same as the shedcling frequency

Two useful expressions result:

1 1 In a real vortex street, T i 0 far downstaream ancl then -4- d S' Or, if SL/d is linown from measurements, then T7/lr0 may be computed.

An example of measured values of L/d is shown in figure 2 1. These were talien from the streamline plot obtained by Kovasznay (ref. 11) a t R=53 (for which S=0.128). There is a little scatter, but l/d does approach the constant value l/S=7.8.

The scatter, while relatively unimportant in the ease of Ijd, gives very low accuracy for values of V/U, calculated from equation (B6). These have also been plottecl in figure 21. I t is surprising that some of tlie values, near the cylinder, are negative (corresponding to values of l/d higher than 118); i t is believed that this results from the combined ciifficulty of estimating the vortex centers, especially near the cylinder, and the sensitivity of equation (B6). (However, i t must be noted that negative values of V are not impossible. Negative V simply means that the vortex velocity is clirected upstream relative to the fluid, while i t is still downstream relative to the cylinder. Such a possibility exists a t low values of zltl, where the mean velocity a t the eclgcs of the wake is consiclerably higher than U,.)

Another way to obtain V/U, is to assume tliat the vortex centers movc with tlie local mean velocity. Kovasznay's paper includes measurements of mean velocity profiles. From his rcsults the mean velocity along the line of vortex

v U* centers U* 112s brcln clclt,c.rmined and from it -=I--

Uo Uo liss been calrulatetl. Tlie result is plotted in figure 21. Near tlic cylintler it docs not agree with the values obtained by the previous metliotl; it is believed that this is principally due to the difficulties mentio~ied above and tliat the deter- mination of V/Uo from 1 - (U*/Uo) is more accurate.

22 REPORT 1 19 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

12

8

+ uo

4

0

-4

4

3

h 7 ,yo

2

I

o 10 20 30 40 50 60 6

FICURE 21.-vortex-street geometry Calculated from da ta 111 reference 11.

LATERAL SPACING

The lateral spacing, a t least initially, must be determined by conditions near the cylinder. The way in which this spacing increases downstream is discussed, for the stable range, in the section "Spread of Vortex Street."

I n the irregular range, the dependence of the shedding frequency on the distance between the free vortex layers, noted by Fage and Johansen (see the section "Shedding Frequency"), leads to an interesting estimate of the initial lateral spacing of the free vortices. The maximum distance d' between the free vortex layers, instead of the cylinder dimension d, may be used to define a new Strouhal number

d' S '=nl - u 0

(B7)

Fage and Johansen found that, whereas S varies considerably with cylinder shape, 8' is practically constant for a variety of bluff cylinders. Kow the initial lateral spacing h' of the free vortices will be roughly the same ss d', possibly a little smaller :

h' -- d' -(I-€) (Bg)

Then, comparing with equations (Be) and (B7),

hf- 1-t -- 1 1-(VIUO)

6' 039)

From the measurements of Fage and Johansen, S'=0.28. 1 - 6

The factor 1 - (V/Uo) =I . Thus equation (B9) gives h'/l=

0.28; that is, the spacing ratio agrees with KbrmAn's value, a t least close to the cylinder.

SHEDDING FREQUENCY

There is yet no adequate theory of the periodic vortex shedding, and i t is not clear what is the principal mechanism which determines the frequency.

The downstream spacing ratio is related to the shedding frequency by equation (B3) and to the lateral spacing by a stability criterion (e. g., KbrmbnJs value of 0.28 for the idealized street). I t might be considered that the shedding frequency is determined by the spacing requirement, or, conversely, that the shedding is primary and determines the downstream spacing. The latter viewpoint seems the more plausible one; that is, the shedding frequency is estab- lished by a mechanism which depends on features other than the vortex spacing. I t is necessary to obtain a better under- standing of the flow field near the cylinder. One of the elements involves the problem of separation, particularly the nonstationary problem. Another that requires more study is the flow field directly behind the cylinder.

With a better knowledge of these, and possibly other, features i t may be possible to set up a model of the shedding mechanism. I n the meantime i t is not clear whether the vortex spacing requirement is decisive in determining the frequency.

DESTABILIZATION a OF SHEDDING

The following experiment illustrates the dependence of the periodic shedding on "communication" between the free vortex layers, that is, on the flow field directly behind the cylinder. A thin flat plate was mounted behind the cyl- inder in the center plane of the wake (fig. 22). It was completely effective in stopping the periodic shedding. Spectrum measurements in the flow on one side of the plate are shown in figure 22. ~t ~ = . 7 , 5 0 0 no significant fie- quencies could be separated out from the continuous back- ground. ~t R=3,200 there were several predominant fie- quencies (all higher than the shedding frequency for the cylinder), but, by the time the flow reached the end of the plate, 5 diameters downstream, i t was completely turbulent. (The shedding frequency nl for the cylinder is marked in the figures.)

The important effect, on the shedding, of the flow field directly behind the cylinder is apparent. Probably an even shorter length of plate would be effective in destablizing the periodic shedding, and there may be a most effective

8 The ctabillty ronsidrled in thls section 3s not a ~ t h recpect to lammar-turbulent transi- tion, it concern? the stab~lity of the periodle shedding (rf. the sectIonL1Stdb~lity")

THE DEVELOPMENT OF TURBULENT WAKES FROM VORTEX STREETS 23

.010 ,0050

.008 ,0040

,006 - .0030 - E G E

.004 .0020

.002 .oo 10

0 I00 200 300 400 0 400 800 1,200 1,600 n n

( a ) R=3,200; z/d=2. (b) R= 7,500; z/d= 1.

FIGURE 22.-Effect o f downstream plate on wake frequencies.

position for such an interference element. Kovasznay remarks that the hot-wire probe used in investigating the vortex street must be inserted from the side, for if i t lies in the plane of the street i t has a strong destablizing influence.

A more complete study of the destabilization of shedding by such interference devices may be quite useful from a practical viewpoint. Structural vibrations and failures are often attributed to the periodic forces set up on members exposed to wind or other flow (smoliestacks, pipe lines, struc- tural columns, to mention a few). I n many cases i t might be possible to destabilize the vortex shedding by addition of simple interference elements or by incorporating them in the original designs. I n the case where one member is buffeted by the wake of another the same principle might be applied.

S P R E A D O F V O R T E X S T R E E T

I t has been observed by most investigators that the spac- ing ratio h/l is Kirm6nJs value (0.25) close to the cylinder but increases rapidly downstream. The increase of h/l is mainly due to the increase of h, since 1 changes very little (fig. 21). In tshe stable range this is the result of viscous diffusion of the real vortices.

Hooker (ref. 28) has made an interesting analysis. First, a real vortex has a core of finite radius; its center is the point of zero velocity and maximum vorticity. Hooker shows that in a vortex street, where the velocity field of the other vortices must be taken into account, the points of zero velocity and maximum vorticity do not coincide. The point of maximum vorticity is unchangect, but the point of zero velocity is farther away from the center of the street. As the vortex decays, the point of zero velocity moves farther out, its distance from the center of the street increas- ing almost linearly wit11 time. Thus the spacing based on vorticity centers remains constant, while the spacing based on velocity centers increases linearly. Hooker's calcula- tion of the linear spread checks fairly well with some pic- tures taken by Richards (ref. 29) in the wake of an elliptical cylinder having a fineness ratio of 6: 1 and the major diameter parallel to the free-stream velocity.

However, the spread of the wake is not always observed to be linear. Among the different investigators there is a, Iargc variation of results, apparently dependent on the experimental arrangement. I n Richards' experiment the cylinder was towed in a water tank and the vortex patterns were observed on the free surface.

I n KovasznayJs experiment the cylinder was mounted in a wind tunnel, the arrangement being similar to the one used here (see the section "Experimental DataJ'). On his plot of the streamlines at R=53 the downstream spread of the vortex street is parabolic rather than linear. I t is possible to fit his results by a somewhat different applica- tion of Hooker's idea, using decaying vortex filaments.

Each vortex in the street is considered to behave like a single vortex filament carried along by the fluid, its decay or diffusion being the same as if it were at rest. The decay of such a vorte; is described by a heat equat,ion, whose solution is (ref. 30, p. 592):

where q is the tangential velocity a t the distance r from the center and at the time t . The circulation is r. This is essentially a vortex with a "solidJJ core and potential outer flow joined by a transition region in which the velocity has a maximum value. This maximum velocity is

and occurs at the radial distance *

r*=2.24(vt)lI2 (B12)

Here r* is defined as the vortex radius. Thus the radius increases as t1J2 and the maximum velocity

decreases as t-'I2. I n the vortex street, the time t is replaced by the downstream distance x. Since the vortices move with the velocity U* rather than U,, the dimensionless time

uo x 8=- - is appropria.te, where U* also varies downstream U* d

(see the section "Downstream Vortex SpacingJ'). When a pattern of such vortices is superimposed on a uni-

form flow, i t is possible to calculate the velocity fluctuation at a point due to the pattern passing over it.

Now the following hypothesis is added. I t is assumed that the vortex radius r* is equal to the width h of the street. Then the width of the street increases as xlJ2.

A second result follows. The maximum velocity fluctua- tion (observed by a hot-wire, say) will occur on the line of cortex centers and will have the amplitude

that is, the hot-wire encounters instantaneous velocities varying from U* (because of vortex centers passing over it) to U*+q* because of the fields of vortices on the other side

24 REPORT 1 19 1-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

of the street). Relations (Bl l ) and (B12) then give the downstream behavior of the maximum fluctuation amplitude.

The results may be summarized as follows: (a) Wake width h = eli2.

(b) The maximum amplitude of fluctuation u* occurs on the line of vortex centers (so there are two maximum points across the wake).

(c) U* (d) u*=0.36 (rl2ah). A comparison of the above predictions was made with

calculations based on Kovasznay's measurements which in- clude profiles of velocity fluctuation amplitude (cf. fig. 11) as well as the streamline plot. The following comparisons were obtained, item by item:

(a') The time variation of hld, determined from the vortex canters on the streamline plot is shown in figure 21. The parabola h/d=0.59(e- 6)'12 is shown for comparison.

(b') The line of maximum velocity fluctuation lies slightly inside the line of vortex centers and is fitted by h*/d=0.53 (e-6)''2.

(c') The time variation of u* is also plotted in figure 21. (Actually Kovasznay's maximum root-mean-square values u', are plotted, but these should cliffer from u* only by a constant factor.) The curve u',/ Uo=0.26(8-6)-'12 is shown for comparison. The points could be fitted better, but the curve was chosen again to have the origin P=6.

(d') A comparison with (d) may be made by estimating the strength r of the vortices. Such a consideration, in fact, led to the present model, for i t was found that the magnitude of the observed velocity fluctuations could be accounted for only by assuming that the radius of the vortex core is about equal to the width of the street. This observation had already been made by Fage and Johansen (ref. 8), for R=2X104. If the free vortex layer is represented by a velocity discontinuity U= U, to U=0, then the circulation is U, per unit length and "the circulation" flows with the velocity U0/2. On the other hand, the rate a t which circu- lation enters one side of the street is nlr , where r is the cir- culation per v o r t e ~ . ~ Therefore

For Kovasznay's example, S = 0.13, so r .x 4 U,d. Then, cornparing with (d), the maximum fluctuation in the initial part of the wake is

the width of the street; if the core is assumed to be much smaller, the calculated velocity fluctuations are much larger than those observed. Ailso, if the cores were very small compared with the width of the wake, four peaks instead of two would be observed in the profile of th? velocity fluctua- tion amplitude.

I REFERENCES

1. Strouhal, V.. Uber eine besondere Art der Tonerregung. Ann. Phys. und Chemie, Neue Folge, Bd. 5, Heft 10, Oct. 1878, pp. 216-251.

2. Rayleigh, Lord. The Theory of Sound. Val. 11. Dover Publica- tions, 1945.

1 3. Rayleigh, Lord. Acoustical Observations. Phi!. Mag., ser. 5, vol. 7, no. 42, Mar. 1879, pp. 149-162.

4. Flachsbart, 0 . : Geschichte der experimentellen Hydro- urid Aero- mechanik, insbesondere der Widerstandsforschung. Bd. 4, Teil 2 der Handbuch der Experimentalphysik, Ludwig Schiller. ed., Akademische Verlags G. m. b. H. (Leipzig), 1932, pp. 3-61.

5. Ahlborn, F.: Uber den &lechanismus des Hydro-dynamischen Widerstandes. Abh. Geb. Naturwiss., Bd. 17, 1902.

6. Benard, H.: Formatio~i de centres de giration l'arrihre d'un ob- stacle en mouvement. Comp. rend., Acad. Sci. (Paris), t. 147, Nov. 9, 1908, pp. 839-842.

7. Von KBrmSn, Th., and Rubach, H.: Uber den Mechanismus des Flussigkeits und L~~ftwiderstandes. Phys. Zs., Bd. 13, Heft 2, Jan. 15, 1912, pp. 49-59.

8. Fage, A., and Johansen, F. C.: On the Flow of Air Behind an Inclined Flat Plate of Infinite Span. It. & M. No. 1104, British A. R. C., 1927; also Proc. Roy. Soc. (London), ser. A, vol. 116, no. 773, Sept. 1, 1927, pp. 170-197.

9. Fage, A,, and Joha~isen, F. C.: The Structure of Vortex Sheets. R. & M. No. 1143, British A. R. C., 1927; also Phil. Mag., ser. 7, vol. 5, no. 28, Feb. 1928, pp. 417-441.

10. Fage, A.: The Air Flow Around a Circular Cylinder in the Region Where the Boundary Layer Separates From the Surface. R. & &f. No. 1179, British A. R. C., 1928.

11. Kovasznay, L. S. G.: Hot-Wire Investigation of the Wake Behind Cylinders a t Low Reytlolds Numbers. Proc. Roy. Sac. (Lon- don), spr. A, vol. 198, no. 1053, Aog. 15, 1949, pp. 174-190.

12. Dryden, Hugh L.: Turbulence Investigations a t the National Bu- reau of Standards. Proc. Fifth Int. Cong. Appl. Mech. (Sept. 1938, Cambridge, Mass.), John Wiley & Sons, Inc., 1939, pp. 362-368.

13. Dryden, Hugh L.: A Review of the Statistical Theory of Turbu- lence. Quart. Appl. Math., vol. I, no. 1, Apr. 1943, pp. 7-42.

14. Leipmann, H. W., Laufer, J., and Liepmann, Kate: On the Spec- trum of Isotropic Turbulence. NACA T N 2473, 1951.

15. Liepmann, H. W.: An Approach to the Buffeting Problem From Turbulence Considerations. Rep. No. SM-13940, Douglas Aircraft Co., Inc., Mar. 13, 1951.

16. Liepmann, Hans Wolfgang, and Laufer, John: Investigations of Free Turbulent Mixing. NACA TN 1257, 1947.

17. Liepmann, H. W., and Robinson, M. S.: Counting Methods and Equipment for Mean-Value Measurements in Turbulence Re- search. NACA T N 3037, 1953.

18. Robinson. M.: Ten-Channel Statistical Analvzer for Use in Tur-

assuming h =d a t this low Reynolds number. The largest value of u,,lUo in K~~~~~~~~~~ example is 0.14 at z,,+7, corresponding to u*/ U , = 0.2. The order of magnitude of this estimate is quite sensitive to the size of the core relative $0

"he velocity at the outer edge of the layer is actually abollt 1.5U,, hut esperiments indicate that only ahout hall the vorticity 01 theshearlagergoes into individual vortices. Therelore, the value U.dI2S is a fair estimate.

bulence Research. Aero. Eng. Thesis, C. I. T., 1952. 19. Fluid Motion Panel of the Aeronautical Research Committee and

Others (S. Goldstein, ed.): Modern Developments in Fluid Dynamics. Vol. 11. The Clarendon Press (Oxford), 1938.

20. Stanton, T. E. , and blarshall, Dorothy: On the Eddy System in the Wake of Flat Circular Plates in Three Dimensional Flow. R. & M. No. 1358, British A. R. C., 1930; also Proc. Roy. Soc. (London), ser. .4, v0l 130, no. 813, Jan. 1, 1931, pp. 295-301.

THE DEVELOPMENT OF TURBULENT WAKES FIXOM VOI~'I'ES B'VI~EETS 25

21. Spivack, Hermann N.: Vortex Frequency and Flow Pattern ill tho Wake of Two Parallel Cylinders a t Varied Spacing Nortnal t.o an Air Stream. Jour. Aero. Sci., vol. 13, no. 6, Juttc 1!)4fi, 1)1,. 289-301.

22. Schiller, L., and Linke, W.: Druck- und Reibungswidcrst,and tlcs Zylinders bei Reynoldsschen Zahlen 5000 bis 40000. Z. 1'. XI . , Jahrg. 24, Nr. 7, Apr. 13, 1933, pp. 193-198. (Availal~lc ill English tra~lslation as NACA T M 715.)

23. Dryden, Hugh L.: The Role of Transition From 1,arninar to Tur- bulent Flow in Fluid Mechanics. Bicentennial Conference, Univ. of Penn., Univ. of Penn Press (Philadelphia), 1941, pp.1-13.

24. Taylor, G. I.: Stability of a Viscous Liquid Contained Between Two Rotating Cylinders. Phil. Trans. Roy. Soc. (London), ser. A, vol. 223, no. 612, Feb. 8, 1923, pp. 289-343.

25. Pai Shih-I: On Turbulent Flow Between Rotatillg Cvlinders. Ph.

26. Sclt~tl)tnt~~r, (;. I % . , tbrtti Skranlsl.atl, [I . I<.: 1,atninar Boundary- 1,nyc.r 0scillat.ions anil Stal~ility of TJatninar Flow. Jour. Aero. Sci., vol. I . & , 1 1 0 . 2, 1''cl). l!I47, pp. A!)--78.

27. rl'o\vt~sl:~~(l, h. h : 121otnc11t~n111 and T':ncrpy Difft~sion in the Tur- 1)11lcnt \trakc of a Cylinder. Proc. Roy. Soc. (I.ondon), ser. A, vol. 1!l7, no. 1048, 3Iay 11, 1949, pp. 124-140.

28. [looker, S. ( i . : On the Action of Viscosity in Increasing the Spac- ing llatio of a Vortex Street. Proc. Roy. Soc. (London), ser. A, vol. 154, no. 881, Mar. 2, 1936, pp. 67-89.

29. Iticllards, G. J.: An Experimental Investigation of the Wake Behind an Elliptic Cylinder. R. & M. No. 1590, British A. R. C., 1933; also Phil. Trans. Roy. Soc. (London), ser. A, vol. 233, no. 726, July 18, 1934, pp. 279-301.

30. Lamb, Horace: Hydrodynamics. Sixth ed., Dover Publications; - . D. Thesis, C. I. T., 1939. I 1945.

TABLE I.-PROBABILITY FUNCTIONS

d l ) ~ ( € 1 Nt I --

2 x keven ......... Random function (Gaussian) e-" K + l ' & -i (1-err<) $20 $r(?) koad

-- ---- - 1 Triangular wave

!Z 1 E .................... v3 IEISJ;

2 4 - -

' Sine wave ............................ - 1- cos-1 E - = I . 312

-- - -- I

Square wave ......................... (,) 0, €>-i Jz

TABLE I1

RING DIMEXSIONS

/ Ring / d . crn / D 1 d . / Did /

TABLE I11

VALUES OF STROUHAL NUMBER FOR VARIOUS TEST REYNOLDS NUMBERS

U. S. G O V E R N M E N T P R I N T I N G O F F I C E : 1955